How to morph / merge / blend from one mesh to another?
I assume you want these two meshes to become one mesh, with a connection of some kind between them. If your meshes have the same number of vertices around their respective outer edges, it's as simple as follows:

First, if you want to keep those edges looking the same when you join them, you need to add a loopcut very close to them, then two more between those two. If you've already done this, or have applied holding edges in another way, feel free to skip this step.

Second, you need to remove the faces at the end of both objects that you wish to join, leaving only that outer edge. This will keep little dimples and grooves from showing up in your final work.

Third, make sure both edges have the same number of vertices. While this isn't strictly necessary for the next operation, it tends to make the mesh look bad, which is what you want to avoid. If they don't match, add loopcuts along either object until the vertex count for the outer edges are the same. (You can check this number without counting in the upper right of blender next to the menu bar. The first number is the number of vertices you have selected, and the second is the number of vertices in your object).

Fourth, hit the space bar and type in 'bridge', then select 'bridge edgeloops'. DON'T CLICK ANYTHING ELSE YET. Hit the 'F6' button on your keyboard. (If you're using a laptop, you may have to hit the 'fn' key simultaneously.) Adjust these settings until you get something you like, hit enter to confirm any typing, and move your mouse out of the menu. You may now move it around as you like and continue with your modeling.